Athens Journal of Business & Economics - Volume 5, Issue 4 – Pages 287-300
https://doi.org/10.30958/ajbe.5-4-2 doi=10.30958/ajbe.5-4-2
The Effect of Allocative Efficiency of Free Markets
on Entropy and its Implications on Taxes
By Stephan Unger*
This article shows that the entropy in a free market is maximized under the allocative
efficiency condition. In contrast to that, it is shown that any pre-determined allocation, as
it is the case with the collection and distribution of taxes, exhibits a higher probability
of minimizing the entropy in the system, where the loss in entropy corresponds to the
deadweight loss caused by the excess burden of taxation. The implications are that
any chaotic system, or exchange economy, converges to an optimal structure of wealth
distribution which maximizes social welfare, in contrast to a randomly, pre-determined
distribution of wealth.
Keywords: Entropy, Social Welfare, Deadweight Loss, Perfect Competition, Allocative
Efficiency, Taxes.
Introduction
Adam Smith argued in his seminal work, "The Wealth of Nations" (1776),
that the invisible hand of free markets, also known as the first welfare theorem,
channels an economic system to reach the correctly required level of production.
This required level of production would lead to a system where all resources are
optimal allocated.
Arrow and Debreu showed mathematically that free markets reach an
equilibrium that is Pareto efficient in allocation (Frank 2003). In the state of a
Pareto efficient allocation the entropy of the system is at its maximum.
The notion of entropy was first defined by Carnot in 1803 (see Oliveira 2014)
but got adapted by Boltzmann (1866), Gibbs (1878) and Maxwell (1871). The
domain of entropy is widely spread among thermodynamics, statistical mechanics,
communications and information theory.
Technically, entropy is an extensive state variable that is definable for any
material substance or any system. The term "extensive" means that it is
proportional to the "size" of the system (like volume or mass) in contrast to an
"intensive" variable (like temperature, pressure or density). The term, along with
the underlying concept, was introduced by Rudolph Clausius, in the 19th century,
to help explain the tendency of temperature, pressure, density and chemical
gradients (in fact, all sorts of gradients) to flatten out and gradually disappear over
time. The physical law behind the concept is deceptively simple to state: If the
system is isolated and closed, so that it does not exchange matter or energy with
any other system, its entropy increases with every physical action or transformation
that occurs inside the system.
*Assistant Professor, Saint Anselm College, USA.
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Entropy can never decrease in an isolated system or in the universe as a
whole. When the isolated system reaches a state of internal equilibrium its entropy
is maximized. When two systems interact with each other, their total combined
entropy also tends to increase over time. This non-decreasing property, roughly
speaking, is known as the Second Law of Thermodynamics, or just the "entropy
law". (Ayres 1996)
In information theory, entropy defines the average amount of information
produced by a system. In terms of an exchange economy, entropy encompasses all
information about all transactions taking place. Considering that any kind of
wealth distribution among all agents in this system is based on this information, a
state of maximum entropy is desirable. Jaynes (1965) established the principle of
maximum entropy which states that "the probability distribution which best
represents the current state of knowledge is the one with largest entropy".
The distribution of wealth in a system in which entropy is not maximized
means that the distribution does not consider all information embedded in the
system, meaning all economic data and variables. Therefore the realized distribution
can also not be optimal. According to the second law of thermodynamics, every
natural system tends to maximize entropy by increasing its complexity. By adding
information to the system, the complexity of the system increases. By sufficient
addition of information it can be shown that the system converges to an equilibrium
which is optimal.
Such collective mechanisms are e.g. given in a free market environment
where each unrestricted transaction taking place serves as an additional input of
information, thus optimizing the distribution of wealth in an economy.
There are two ways to determine the collection and distribution of wealth in
an economy: Through a social market economy or a free market. The existence of
a centralized entity in a social market economy is sufficient but not necessary as it
might administer the actual distribution of wealth. Concurrently it is highly
inefficient to have the centralized entity decide about the shape of wealth
distribution as the probability that this determination is optimal converges to zero
with increasing amounts of trials to realize an optimal distribution of wealth.
In contrast to that, a determination of wealth by a free market is based on the
system's agent's learning progress through trial and error by the implicit generation
of information in every iteration. Robinson (1969) suggests that the seller may be
conceived to equate marginal revenue to marginal cost “either by estimating the
demand price and the cost of various outputs, or by a process of trial and error”.
Therefore learning outcome is evaluated at each distribution step in terms of
realized entropy. The probability that the system reaches an optimal status, with
respect to its maximum entropy, increases as the number of trials increases as
information . This paper shows how the maximization of entropy leads to
an optimal distribution of wealth.
I will first give an overview of wealth distribution in a social market
environment vs. a free market environment. Then we take a look at the starting
conditions necessary to determine the evolution of a tax collection regime with
subsequent redistribution of wealth vs. the consolidation of a wealth distribution in
a free market.
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After explaining the difference of wealth distribution evolution in both
systems, I will explain the concept of entropy maximization in an exchange
economy, showing that the quality of a certain realized wealth distribution can be
measured by the Kullback-Leibler divergence, even if the optimal distribution is
unknown.
Finally, I will analyze centralized taxation and its implications on the
deadweight loss, as well as its meaning to entropy in an exchange system.
Additionally I will briefly explain the effect of entropy on monopolies.
Wealth Distribution in a Social Market vs. Free Market
"Prior to the passage of the 16th Amendment in 1913, the United States
government funded its operations mainly through excise taxes, tariffs, customs
duties and public land sales. The federal government had relatively few expenses
compared to today and did not have as much need to raise large amounts of
money. The U.S. government was capable of paying for its expenses without an
income tax prior to 1913 largely because it had fewer responsibilities. Thomas
Eddlem noted in "The New American" that the federal government's
responsibilities were limited to basic operational matters and did not include such
modern expenses as social insurance programs, welfare programs or agricultural
subsidies." (Gabriel 2018)
Article I, Section 9 of the U.S. Constitution states, "No Capitation, or other
direct, Tax shall be laid, unless in Proportion to the Census or enumeration herein
before directed to be taken." A federal income tax had been enacted in 1861, but
was struck down as unconstitutional by the Supreme Court in 1895 because it was
found to be a direct tax outside the constitutional constraints. Congress removed
these limits in 1913 with the passage of the 16th Amendment, which allows it to
impose income taxes specifically, "without regard to any census or enumeration."
The history shows that a social system is able to function without the
imposition of personal income taxes. Raising taxes on income lowers purchasing
power and the allocation ability. If no taxes were imposed, an agent could
determine his own allocation and spending amount towards system relevant
factors such as energy, health, social security or pensions.
On the supply side, every agent would observe the realized spending
distribution and prices would be set according to the willingness of providing the
services requested. The probability that both agents don't equilibrate is initially
very high, thus information entropy is very low in this system, because none of the
agents knows the other agents adjustment capability.
Negotiation among both agents might lead to a market clearing price. This
price might be an exchange price for two commodities, but might also be the
exchange between one unit of labor and a commodity. For the system, information
entropy would increase. Considering many agents, entropy is very small at this
stage with just two agents contributing towards an n-agent's system entropy with
just one cleared market price.
The process of finding a market equilibrium in which some services clear will
accelerate as soon as more and more information is revealed on the supply as well
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as on the demand side, thus entropy increases as the complexity increases, because
more and more transaction information has to be considered. As entropy increases,
resources are allocated more and more efficient towards relevant and market
clearing sectors. All sectors start getting priced fairly which means that wealth
starts getting distributed optimally. Moreover, economies of scale will lead to fair
pricing, e.g. electricity provider will be able to produce at rates which consumers
will be willing to accept.
For a market clearing system with maximum entropy, the probability that the
wealth is distributed optimally converges to 1. Therefore, a system which pre-
determines the shape of the distribution of the resources just realizes one possible
outcome of an N-trial entropy-maximizing system. The probability that this
outcome is optimal vanishes with increasing number of trials.
Prerequisites
Social Market System
First we need to define the starting conditions for the two systems subject to
comparison: The social market exchange economy and the free market economy.
In order to run an N-trial exchange system the starting values are set randomly in
both systems.
For the social market economy we can assume that a central entity chooses
with a certain probability the distribution of wealth W, conditional on the tax
collection distribution T, based on available market information such that
, (1)
provided that , is hard to evaluate since in a social
market environment, the distribution collection of taxes given a certain wealth
distribution is exposed and subject to political debate and choice. Therefore we
can interpret as measure for redistribution of wealth. is the
probability of realization of a certain wealth distribution and is the
probability of a certain tax collection choice.
The tax brackets are subject to change, based on constant updates. The
problem a social market system faces is that with increasing number of updates (or
iterations), the complexity of the system increases too. This makes it impossible
for a central entity, besides political pressure, to update its distribution conditional
on all information.
Moreover, the social market system faces this problem in two ways: First, in
the determination of the tax collection brackets, and second in the choice of
redistribution of the wealth. As we will see, the problem of system complexity
turns out to be crucial when it comes to the determination of a socially optimal
distribution of wealth.
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Free Market System
A fundamental requirement for a free market to exist is pure competition.
There is general agreement upon concrete circumstances which are prerequisite to
the functioning of pure competition. Professor Chamberlin asks: (1) that the
number of buyers and sellers be large “so that the influence of any one or several
in combination is negligible”; (2) and that an identical good be offered by all the
sellers. (Triffin 1947c)
Perfection of the market is identified with the identity of the product. In a free
market environment the starting condition is set by buyers and sellers who
negotiate about a transaction price. Their negotiation may also be based on given
market information, e.g. cost of resource, cost of labor, comparable goods, etc. By
realization of transaction prices , the distribution W is formed successively,
where each new transaction price, displays new information, serving as input for
the generation of subsequent transaction prices. Thus, the distribution of wealth in
a free market environment is based on realized transaction prices:
, (2)
provided that , is generated successively, simply by
realization of transaction prices based on the given distribution of wealth, where
the starting distribution of wealth might not be equal among the agents due to
different endowments of resources, etc.
Since T and A are functions of N iteration steps, both systems might face the
same starting conditions, but as we will see, will develop in a different way over
time, due to different complexity accumulation.
Entropy Maximization in an Exchange Economy
Assuming that we denote the unknown optimal distribution of wealth by X
and the approximating distribution, either set by a government, as or determined
by the free market, as . To determine which distribution approximates better X,
we can measure the Kullback-Leibler (KL) divergence. Since the distribution with
the least distance to the real distribution is the preferable one, we need to minimize
the KL-divergence. By minimizing the KL-divergence between X and ,
respectively X and we automatically maximize the entropy of the system as
, (1)
, (2)
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where X is distributed as , as g(x) and as f(x). For large N we can find
the best parameters, which minimize the KL-divergence for both approximating
distributions:
)
, (3)
)
, (4)
where ) is constant.
Therefore the KL-divergence gets minimal if the terms )
and , defining the entropy, get maximized. That means
that with increasing number of N transaction prices, the KL-divergence converges
to its min.
Thus, a free market environment can always generate and encompass more
information than any fixed chosen distribution. This leads to the conclusion that a
free market framework approximates more accurate any optimal distribution of
wealth than a chosen distribution. The reason lies in the fact that any central entity
which chooses a certain distribution can only consider a certain amount of
complexity of the system, and is thus limited in gathering all market information.
Of course this condition holds just if the rate at which the complexity of the system
increases is below the rate at which information is generated.
Efficient Allocation
To prevent a system without tax payments from a collapse due to social unrest
the agents will have to self-organize by efficient allocation of their resources.
Since the system consists of learning agents, the system will yield for an
equilibrium with probability , which is Pareto-optimal for all system entities.
For production to happen at marginal cost and equalizing marginal benefit of
the consumers, all information has to be known (meaning that the system
complexity is at its maximum), thus entropy has to be maximized.
In order to ensure that the system is allocating its resources efficiently by
maximizing its entropy it has to satisfy following Shore and Johnson axioms:
The maximum has to be unique.
Predictions about the resource allocation should be coordinate invariant.
Subset independence: If the probability or realization of a resource
allocation path is increased and the probability or realization of
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another resource allocation path is decreased then no other realizations
are affected.
System independence: The predicted joint probability of realizing two
resource allocations must be the product of the predicted marginal
probabilities.
Efficient allocation prevails when the marginal cost equal marginal revenue,
thus when the quantities are exchanged at an efficient price . Both functions,
g(x) and f(x), try to approximate these efficient quantities and prices.
In order to measure if g(x) or f(x) better approximates X, we have to take the
model which maximizes the likelihood function .
By using the Bayesian Information Criterion (BIC) we can determine which
distribution approximates better the optimal distribution:
. (5)
A small BIC is preferred, therefore we are interested in evaluating the
performance of g(x) and f(x) when and how the BIC changes:
, (6)
(7)
We can see that the rate at which the BIC information criterion gets minimal
is when . Since g(x) is fixed, whereas f(x) is not fixed, thus allows
.
The rate at which the BIC approximately changes is:
, (8)
, (9)
, (10)
. (11)
A system without a central entity determining how many resources are levied
and subsequently allocated is automatically not able to maximize its entropy.
In such a system, the entropy is the average information available to all
entities which are part of the system:
. (12)
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In a system where a central entity determines the resource allocation, the
resource allocation is a deterministic function, or in other words, the probability
for a random allocation to a certain project to be optimal is zero:
If , then since
(13)
The result implies that the probability for an infinite set of trials to minimize
the entropy of the exchange system is higher than for just one pre-determined
distribution of allocable resources.
This implies that resource allocation is only efficient in free markets.
Any pre-determined resource allocation is just one possible outcome of any
randomly generated resource allocation. Equipping a central agency with the
power to impose and levy taxes means adding constraints to the system. If every
entity in the system is free to choose its optimal resource allocation with respect to
its individual risk preference and utility function, then the aggregate utility is
Pareto-optimal. This implies also that such a system would be socially sustainable.
Deadweight Loss
The deadweight loss (DWL) is defined as the loss in welfare by realization of
an equilibrium in an exchange economy under imposed taxes. The socially
optimal equilibrium should prevail where marginal cost of production equal the
marginal cost of benefit, or when supply meets demand. By the imposition of
taxes, the market is just able to realize quantities less than it could produce in the
optimum and faces a reduction in the marginal benefit as realized prices for
demand side are higher and on the supply side are lower than in the social
optimum. The implications of this loss in welfare are well known (Hausman 1981,
Case 1999, Hines 1999, Lind and Granqvist 2010). What is interesting is how the
rate of complexity reduction, and thus the loss in entropy in a system with a fixed
choice of wealth distribution, can explain the size of the deadweight loss:
(14)
, (15)
, (16)
(17)
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The welfare loss caused by taxation can therefore be explained by the rate at
which information cannot further contribute towards the complexity of the system
caused by a determined tax brackets and wealth distribution. The difference
between a social market entropy and a free market entropy could be interpreted as
the opportunity costs of information.
An increase in information, as given in a free markets setup, maximizes the
complexity, thus the entropy of the system, and yields to an optimal wealth
distribution. But since the complexity of the system increases not to the power of n
(the number of information bits), the BIC information criterion tends to get
minimal as , where the BIC information criterion measures the Kullback-
Leibler (KL) divergence between an unknown social optimal wealth distribution
and a realized wealth distribution. The reason for the superiority of free markets
vs. a pre-determined wealth distribution can be explained by the phenomenon that
the increase in information entropy of the system leads to allocative efficiency in
the market, which minimizes the KL-divergence, as can be shown and measured
by the BIC.
The rate at which the optimal wealth distribution is approximated,
respectively the pre-determined wealth distribution is outperformed, corresponds
to the rate at which information is added to the system, namely
, (18)
where i is a welfare-state and is the probability of x given i as the previous
welfare-state with a given wealth distribution.
Allocation under a Monopoly
One of the critics of free markets is that they tend to form monopolies that
allocate resources non-optimally. Monopoly exists for Mises (1963) when "…the
whole supply of the commodity is controlled by a single seller or a group of sellers
acting in concert". He further states that if monopoly prices do exist, then they are
an "infringement of the supremacy of the consumers and the democracy of the
market". Mises also argues that although most monopolies and monopoly prices
are made possible by government intervention in the free market (tariffs, licenses,
etc.), there are certain instances in which monopoly (and monopoly prices) arise in
the unhampered market. He specifically mentions natural resource monopoly,
geographic monopoly, limited-space monopoly, and monopoly that might arise
because consumers place a "special confidence…on the individual or firm con-
cerned on account of previous experience," as with certain trademarked drugs.
But for Rothbard (1962) there is no social "problem" associated with
monopoly in a free market. Monopoly prices cannot be defined logically, let alone
established in a free market. According to Kirzner (1973), if markets are always
competitive so long as there is freedom to buy and sell, then in a free market there
is always competition and never any monopoly.
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Paretian analysis defines competition as incomplete when marginal costs are
increasing and as complete when they are decreasing. When costs are decreasing
the rules of profit maximization induce the firm, not to limit its production at a
certain level, but to expand it indefinitely. In the opposite case, when increasing
marginal costs limit the size of the profit maximizing output, Pareto considers that
competition is “incomplete”: Each producer finds it advantageous to limit his
supply. (Triffin 1947a)
In the case of , under independent trials, we can assume that provided
information leads to an efficient usage of resources and therefore to cost reduction
in a long term. Relating the existence of monopolies to the reduction of entropy in
a system we can measure the loss of entropy again at the same rate as the loss of
entropy in the case of the dead weight loss.
An interesting point arises when we consider interdependence in a free market
environment: Especially the interdependence of buyers and sellers. When a
commodity is sold, sellers and buyers are linked in an immediate way since the
price received by the first is disbursed by the latter. (Triffin 1947b) Interdependence
between buyers and sellers might arise a bilateral monopoly, which in turn would
imply that free markets automatically cause monopolies, which would mean that
even in a free markets environment, entropy could not be maximized.
But according to Mises (1963) "monopoly prices emerge as differentiated
from competitive prices" could only evolve if the demand for the product is
inelastic. Hence, it is not "monopoly" as such that is catallactically relevant for
Mises, but only the "configuration" of the demand function and the emergence of
monopoly prices. (Mises.org 2005)
Thus, it is important to take a look on the demand side, meaning, if demand
stays elastic, entropy can still be maximized even under monopolistic market
conditions.
Monopolies don‟t tend to exist for a long time if exposed to a free market
environment, even if entry-levels are high. Events like technology disruptions or
evolutionary change in society and methods always threaten monopoly powers as
they evolve out of nowhere and might be able to satisfy demand which was not
even there before. Furthermore, it is important to remark that industries which
have a higher rate of monopoly formation and longer persistence of monopoly
power are those industries in which government has a large hand in regulation.
One may argue that monopolies don‟t even exist if there is no intervention
into the market over a certain period of time, as there always exists a competitor in
the market. It may only look like a monopoly only under myopically static
analysis. Any broader definition of an industry exhibits an increased level of
competition. Just narrow enough definitions of brand name products appear to
have monopoly characteristics. In a competitive environment and without any
governmental intervention, a company would not be able to set a price at a higher
level and maintain it for a long period of time since consumers would create
demand for products with prices well below the monopoly price. Thus, any
realized dead weight loss by a monopoly set price would not prevail and increase
entropy over time at the rate (17).
The same holds for subsidized monopolies. Since subsidies are a realization
of a conditional probability of attaining a certain distribution, based on available
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market data at a time t, they realize and maintain the dead weight loss by not
allowing entropy to increase. Due to the limitation of incorporated market
information, entropy is capped at a certain level.
In a free market, subsidized monopolies would not exists since there would
not exist any kind of subsidy. The subsidy is nothing else than the realization of
the dead weight loss, caused by conditional redistribution. It is interesting to note
that even subsidized monopolies are challenged by competition. A good example
is given by the digital revolution which challenges even subsidized monopolies
such as the electricity sector, e.g. by the evolvement of the blockchain technology
and the shift towards de-centralized payment systems and residential prosumers.
By jumps in technology, surge in innovation and changes in the social mindset it
gets hard for subsidized monopolies to justify their subsidy. This shows again that,
even if a company appears to be a monopoly, it won‟t be able to maintain its status
as such over a longer period of time without governmental intervention. Exposure
to a free market will have a disruptive effect to any monopoly as entropy
maximization is the natural process, which breaks up any monopoly structures
over time.
Conclusion
This article stresses the effect of wealth collection and distribution of
allocable resources in a social market economy setting in contrast to a free market
system. If no constraints are imposed on a system such as a free market economy,
it will converge to an equilibrium at which all resources are optimally allocated.
This equilibrium can only be realized if the entropy of the system is maximized,
meaning that all offered and asked prices as well as transaction prices and
quantities subject to the exchange are known and therefore contribute to a
maximum complexity of the system.
The maximization of the entropy of a system leads to a convergence of all
possible allocation distributions to an optimal distribution which is allocative fully
efficient. The convergence process is characterized by an information increase of
size N, as well as an N-trial learning process of the system. By identification of the
level of entropy, the agents in the system learn in an iterative process how to
allocate all available resources such that the entropy of the system is maximized.
By maximizing the entropy, the agents automatically maximize the allocative
efficiency of the system, which is Pareto-optimal.
In contrast to a free markets setup, any pre-determination of resource
allocation, as done in a social market economy, restricts the information generation
to let , and thus to maximize the system's entropy. The probability that this
specific choice of distribution is maximizing the entropy of the system converges
to zero when compared to the multiple adaption process of a free market system
which converges to an optimal distribution with certainty.
The Kullback-Leibler divergence allows to measure the distance of a pre-
determined allocation from its optimal one, even if the optimal distribution is
unknown. The relative distance to its optimum is compared to the relative distance
of free market setup. The goal is to minimize the BIC information criterion, thus
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minimizing the KL-divergence. Minimizing the KL-divergence means maximizing
the systems entropy in terms of its complexity. This can only be achieved by
increasing information, which in turn leads to the maximization of entropy.
The probability that imposed taxes on resources for allocating them according
to a pre-determined distribution is optimal, converges to zero, compared to the free
market setup where information is generated and used to minimize the entropy.
Further, it is shown that the rate of social welfare loss from mis-allocated
resources, caused by imposed taxes, equals the rate of accumulating the dead-
weight loss which occurs in a distorted economy.
The result implies that levying taxes reduces social welfare. This means that
the realization of an inefficient allocation of resources leads to a lower level of
living standard.
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